Shape preserving approximations by polynomials and splines

Computers and Mathematics with Applications

Volumn 33, Issue 11, 1997, Pages 59-97

  Kocic L M ^a   Milovanovic G V ^a  

^a UNIVERSITY OF NI   (Serbia)

Author keywords

Approximation of discrete data; Approximation of univariate functions; Convexity; Generalized convexity; Lipschitz property; Monotonicity; Polynomial; Positivity; Shape preserving approximation; Shape preserving interpolation; Spline; Unimodality

Indexed keywords

ALGORITHMS; COMPUTATIONAL GEOMETRY; FUNCTIONS; INTERPOLATION; NUMERICAL METHODS; POLYNOMIALS;

LIPCSHITZ PROPERTY; SHAPE PRESERVING APPROXIMATION; SHAPE PRESERVING INTERPOLATION;

APPROXIMATION THEORY;

EID: 0031167816     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0898-1221(97)00087-4     Document Type: Article

References (165)

1. W. Wolibner, Sur un polynôme d'interpolation, Colloq. Math. 2, 136-137, (1951).
2. W.J. Kammerer, Polynomial approximations to finitely oscillating functions, Math. Comp. 15, 115-119, (1961).
3. S.W. Young, Piecewise monotone polynomial interpolation, Bull. Amer. Math. Soc. 73, 642-643, (1967).
4. F. Deutsch and P.D. Morris, On simultaneous approximation and interpolation which preserves the norm, Bull. Amer. Math. Soc. 75, 812-814, (1969).
5. S. Paszkovski, On approximation with nodes, Rozprawy Mat. 14, 1-61, (1957).
6. I. Mircea, Sur un théorème de W. Wolibner, Anal. Numér. Théor. Approx. 11, 81-87, (1982).
7. Z. Rubinstein, On polynomial δ-type functions and approximation by monotonic polynomials, J. Approx. Theory 3, 1-6, (1970).
8. E. Passow and L. Raymon, The degree of piecewise monotone interpolation, Proc. Amer. Math. Soc. 48, 409-412, (1975).
9. E. Passow, An improved estimate of the degree of monotone interpolation, J. Approx. Theory 17, 115-118, (1976).
10. M.G. Nikolcheva, Parametricheskoe priblizhenie funkcii i monotonnaya interpolyaciya, In The Theory of Approximation of Functions, Proc. Int. Conf. Kaluga 1975, Nauka, Moscow, (Edited by S.B. Stechkin and S.A. Telyakovskiǐ), pp. 272-275, (1977).
11. C. de Boor and B. Swartz, Piecewise monotone interpolation, J. Approx. Theory 21, 411-422, (1977).
12. E. Passow and J.A. Roulier, Monotone and convex spline interpolation, SIAM J. Numer. Anal. 14, 904-909, (1977).
13. C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Volume 27, Springer-Verlag, New York, (1978).
14. L.L. Schumaker, Spline Functions: Basic Theory, John Willey, New York, (1981).
15. F.N. Fritsch and R.E. Carlson, Monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 17, 238-246, (1980).
16. C. Manni, Sull'esistenza di funzioni spline interpolanti monotone, Bollettino U. M. I. 6 (5-A), 59-68, (1986).
17. J.C. Feguson and K. Miller, Characterization of shape in a class of third degree algebraic curves, TRW Report 5322-3-5, (1969).
18. R.E. Carlson and F.N. Fritsch, Monotone piecewise bicubic interpolation, SIAM J. Numer. Anal. 22, 386-400, (1985).
19. R.L. Daugherty, A. Edelman and J.M. Hyman, Nonnegativity-, monotonicity, or convexity-preserving cubic and quintic Hermite interpolation, Math. Comp. 52, 471-494, (1989).
20. A. Edelman and C.A. Micchelli, Admissible slopes for monotone and convex interpolation, Numer. Math. 51, 441-458, (1987).
21. J.R. Rice, Numerical Methods, Software and Analysis, McGraw-Hill, (1983).
22. H. Akima, A new method of interpolation and smooth curve fitting based on local procedures, J. Assoc. Comput. Mach. 17, 589-602, (1970).
23. J.M. Hyman, Accurate monotonicity preserving cubic interpolation, SIAM J. Sci. Statist. Comput. 4, 645-654, (1983).
24. R.K. Beatson and H. Wolkowicz, Post-processing piecewise cubics for monotonicity, SIAM J. Numer. Anal. 26, 480-502, (1989).
25. S.C. Eisenstat, K.R. Jackson and J.W. Lewis, The order of monotone piecewise cubic interpolation, Tech. Report 207, (Edited by Computer Science Department), Yale University, (1981).
26. S.C. Eisenstat, K.R. Jackson and J.W. Lewis, The order of monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 22, 1220-1237, (1985).
27. F.N. Fritsch and J. Butland, A method for constructing local monotone piecewise cubic interpolants, SIAM J. Sci. Statist. Comput. 5, 300-304, (1984).
28. H.T. Huynh, Accurate monotone cubic interpolation, SIAM J. Numer. Anal. 30, 57-100, (1993).
29. Z. Yan, Piecewise cubic curve fitting algorithm, Math. Comp. 49, 203-213, (1987).
30. Lj.M. Kocić and N.V. Blagojević, Adaptive monotone interpolation, In VIII Conference on Applied Mathematics, Univ. of Montenegro, Podgorica, (Edited by M. Jaćimović), pp. 87-94, (1994).
31. G. Farin, Curves and surfaces for computer aided geometric design, In A Practical Guide, Academic Press, (1988).
32. M.L. Varas, A modified duel algorithm for the computation of the monotone cubic spline, In Approx. Theory V, (Proc. Symp. College Station, 1986), pp. 607-610, Academic Press, (1986).
33. P.J.Y. Wong and R.P. Agarwal, Explicit error estimates for quintic and biquintic spline interpolation, Computers Math. Applic. 18 (8), 701-722, (1989).
34. H. Dauner and C.H. Reinsch, An analysis of two algorithms for shape-preserving cubic spline interpolation, IMA J. Numer. Anal. 9, 299-314, (1989).
35. A.W. Roberts and D.E. Varberg, Convex Functions, Academic Press, New York, (1973).
36. E. Neuman, Uniform approximation by some Hermite interpolating splines, J. Comput. Appl. Math. 4, 7-9, (1978).
37. J.A. Roulier, Polynomials of best approximation which are monotone, J. Approx. Theory 9, 212-217, (1973).
38. E. Passow and J.A. Roulier, Negative theorems on convex approximation, In Approx. Theory, (Proc. Int. Symp. Austin, TX, 1976), pp. 509-512, New York, (1976).
39. D.-C. Myres and J.A. Roulier, Markov-type inequalities and the degree of convex spline interpolation, J. Approx. Theory 28, 267-272, (1980).
40. K.G. Ivanov, Shape-preserving interpolation and approximation, Serdica 11, 398-413, (1985).
41. D.F. McAllister, E. Passow and J.A. Roulier, Algorithms for computing shape preserving spline interpolants to data, Math. Comp. 31, 717-725, (1977).
42. J.M. Hyman, Accurate convexity-preserving cubic interpolation, Los Alamos Scientific Laboratory document, LA-UR-80-3700, Los Alamos, NM, (Nov. 1980).
43. E. Passow, Monotone quadratic spline interpolation, J. Approx. Theory 19, 123-134, (1977).
44. J.W. Schmidt, 1-D shape preserving spline interpolation: Existence theorems and construction of optimal splines, Techn. Rep. 07-01-87, Extended version of lectures given at the Banach Center Warsaw, Spring 1986, Techn. Univ. Dresden, Dresden, (1987).
45. J.W. Schmidt and W. Hess, Quadratic and related exponential splines in shape preserving interpolation, J. Comput. Appl. Math. 15, (1987).
46. J.W. Schmidt, Convex interval interpolation with cubic splines, BIT 26, 377-387, (1986).
47. L.L. Schumaker, On shape preserving quadratic spline interpolation, SIAM J. Numer. Anal. 20, 854-864, (1983).
48. A. Lahtinen, On the construction of monotony preserving taper curves, Acta For. Fennica 203, 1-34, (1988).
49. A. Lahtinen, Shape preserving interpolation by quadratic splines, J. Comput. Appl. Math. 29, 15-24, (1990).
50. D.F. McAllister and J.A. Roulier, Interpolation by convex quadratic splines, Math. Comp. 32, 1154-1162, (1978).
51. E. Passow and J.A. Roulier, Shape preserving spline interpolation, In Approx. Theory, (Proc. Int. Symp. Austin, TX, 1976), pp. 503-507, New York, (1976).
52. J.A. Roulier and D.F. McAllister, Approximation by convex quadratic splines, In Approximation Theory III, pp. 757-761, Academic Press, New York, (1980).
53. P. Costantini, On monotone and convex spline interpolation, Math. Comp. 46, 203-214, (1986).
54. P. Costantini, Co-monotone interpolating splines of arbitrary degree. A local approach, SIAM J. Sci. Statist. Comput. 8, 1026-1034, (1987).
55. P. Costantini, An algorithm for computing shape-preserving interpolating splines of arbitrary degree, J. Comput. Appl. Math. 22, 89-136, (1988).
56. D.F. McAllister and J.A. Roulier, An algorithm for computing a shape-preserving osculatory quadratic spline, ACM Trans. Math. Software 7, 331-374, (1981).
57. P. Costantini and R.E. Morandi, Monotone and convex cubic spline interpolation, Calcolo 21, 281-294, (1984).
58. P. Costantini and R.E. Morandi, An algorithm for computing shape preserving cubic spline interpolation to data, Calcolo 21, 295-305, (1984).
59. C. Conti and R. Morandi, Parallel shape-preserving interpolation via Bézier curves, Supercomputer 10, (1994) (to appear).
60. 1 shape-preserving Hermite interpolation, Computing (to appear).
61. P. Costantini, A general method for constrained curves with boundary conditions, In Multivariate Approximation: From CAGD to Wavelets, (Edited by K. Jetter and F.I. Utreras), World Scientific, Singapore, (1993).
62. P. Costantini, Boundary valued shape-preserving interpolating splines, ACM Trans. Math. Software (to appear).
63. 1 Comonotone Hermite interpolation via parametric cubics, J. Comput. Appl. Math. (to appear).
64. V.L. Miroshnichenko, Convex and monotone spline interpolation, In Constructive Theory of Functions '84, Proc. Int. Conf. Constructive Theory of Functions, Varna 1984, (Edited by B. Sendov et al.), pp. 610-620, Publishing House of the Bulgarian Academy of Sciences, Sofia, (1984).
65. 2 conservant la forme, C.R. Acad. Sci. Paris 307, 133-138, (1988).
66. I. Oprea, Spline approximation functions with imposed shape, Prepr. Babes-Bolyai Univ. Fac-Math. Res. Semin. 7, 201-206, (1986).
67. T. Wen, A keep-convex interpolation for monotonic smooth functions, Math. Numer. Sinica 2, 299-306, (1980).
68. P. Dierckx, Algorithm/Algorithmus 42. An algorithm for cubic spline fitting with convexity constraints, Computing 24, 349-371, (1980).
69. D.G. Schweikart, An interpolating curve using a spline in tension, J. Math. Physics 45, 312-317, (1966).
70. H. Späth, Spline Algorithms for Curves and Surfaces, Utilitas Mathematica, Winnipeg, Manitoba, (1974).
71. R.J. Renka, Interpolatory tension splines with automatic selection of tension factors, SIAM J. Sci. Statist. Comput. 8, 393-415, (1987).
72. N.S. Sapidis and P.D. Kaklis, An algorithm for constructing convexity and monotonicity-preserving splines in tension, Comput. Aided Geom. Design 5, 127-137, (1988).
73. N.S. Sapidis, P.D. Kaklis and T.A. Loukakis, A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation, Numer. Approx. 54, 179-192, (1988).
74. J. Kozak, Shape preserving approximation, Computers in Industry 7, 435-440, (1986).
75. Y.Y. Feng and J. Kozak, An approach to the interpolation of uniformly spaced data, J. Comput. Appl. Math. 23, 169-178, (1988).
76. B.J. McCartin, Theory of exponential splines, J. Approx. Theory 66, 1-23, (1991).
77. B.A, Barsky, Exponential and polynomial methods for applying tension to an interpolating spline curve, Computer Vision, Graphics and Image Processing 27, 1-18, (1984).
78. T.A. Foley, Interpolation with interval and point tension controls using cubic weighted ν-splines, ACM Trans. Math. Soft. 13, 68-96, (1987).
79. G.M. Nielson, Some piecewise polynomial alternatives to splines under tension, In Computer Aided Geometric Design, (Edited by R.E. Barnhill and R.F. Riesenfeld), pp. 209-235, Academic Press, (1974).
80. S. Pruess, Alternatives to the exponential spline in tension, Math. Comp. 33, 1273-1281, (1979).
81. P.D. Kaklis and D.G. Pandelis, Convexity-preserving polynomial splines of non-uniform degree, IMA J. Numer. Anal. 10, 223-234, (1990).
82. E. Neuman, Shape preserving interpolation by polynomial splines, Report Nr N 122, Institute of Computer Science, Univ. Wroclaw, Wroclaw, Poland, (May 1982).
83. E. Neuman, Convex interpolating splines of arbitrary degree, BIT 22, 331-338, (1982).
84. R. Delbourgo and J.A. Gregory, Shape preserving piecewise rational interpolation, SIAM J. Sci. Statist. Comput. 6, 967-976, (1985).
85. J.A. Gregory, Shape preserving rational spline interpolation, In Rational Approximation and Interpolation, (Edited by Graves-Morris, Saff and Varga), pp. 431-441, Springer-Verlag, (1984).
86. 2 rational interpolants in tension, SIAM J. Numer. Anal. 30, 595-607, (1993).
87. J.A. Gregory, Shape preserving spline interpolation, Comput. Aided Design 18, 53-57, (1986).
88. M. Sarfraz, Rational interpolation preserving the monotonic shape of the data, Facta Univ. (Niš) 8, 87-95, (1993).
89. V.L. Miroshnichenko, Interpolation of functions with large gradients, In Methods of Approximation and Interpolation, (in Russian), pp. 98-107, Novosibirsk, (1980).
90. Yu.S. Zav'jalov, B.I. Kvasov and V.L. Miroshnichenko, Methods of Spline Functions, (in Russian), Nauka, Moscow, (1980).
91. C.E. Frost and G.L. Kinzel, An automatic adjustment procedure for rational splines, Comput. Graph., 171-176, (1982).
92. J.W. Schmidt and W. Hess, Positive interpolation with rational quadratic splines, Computing 38, 261-267, (1987).
93. J. Pal, Approksimation af konvekse Funktioner ved konvekse Polynomier, Mat. Tidsskrift B, 60-65, (1925).
94. T. Popoviciu, Les Fonctions Convexes, Exposés sur la théorie des fonctions, Hermann and C., Éditeurs, Paris, (1944).
95. G.G. Lorentz, Bernstein Polynomials, Univ. Toronto Press, Toronto, Ont., (1953).
96. W. Wegmüller, Ausgleichung Lursch Bernstein-Polynome, Mitt. Verein. Schweiz. Versich, -Math. 36, 15-59, (1938).
97. H. Berens and R. DeVore, A characterization of Bernstein polynomials, In Approximation Theory III, (Proc. Int. Sympos. Austin 1980), (Edited by E.W. Cheney), pp. 213-219, Academic Press, New York, (1980).
98. R.A. DeVore, The approximation of continuous functions by positive linear operators, Lecture Notes in Math, Volume 293, Springer-Verlag, Berlin, (1972).
99. T. Popoviciu, Sur la conservation de l'allure de convexité d'une fonction par ses polynomes d'interpolation, Mathematica (Cluj) 3, 311-329, (1961).
100. J. Tzimbalario, Approximation of functions by convexity preserving continuous linear operators, Proc. Amer. Math. Soc. 53, 129-132, (1975).
101. T. Popoviciu, Notes sur les fonctions convexes d'ordre supérieur (IX), Bull. Math. de la Soc. Roum. des Sciences 43, 85-141, (1942).
102. P.M. Vasić and I.B. Lacković, Notes on convex functions II: On continuous linear operators defined on a cone of convex functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. Fis. Fiz. 602-633, 53-59, (1978).
103. Lj.M. Kocić, Continuous liner operators defined on the cone of functions convex with respect to Chebyshev system, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. Fis. 735-762, 32-34, (1982).
104. Lj.M. Kocić, Approximation theorems for functions convex with respect to Chebyshev system, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. Fis. 2, 3-10, (1991).
105. Lj.M. Kocić and I.B. Lacković, Generalized monotonicity and approximations by linear operators, In Numerical Methods and Approximation Theory II, (Edited by D. Herceg), pp. 1-6, Univ. of Novi Sad, Novi Sad, (1985).
106. S. Karlin, Total positivity and convexity preserving transformations, In Convexity, Vol. VII, pp. 329-347, Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, (1963).
107. S. Karlin, Total Positivity, Volume I, Stanford Univ. Press, Stanford, CA, (1963).
108. P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, One-Dimensional Theory, Volume 1, Birkhäuser Verlag, Basel, (1971).
109. A. Lupaş, Some prorerties of the linear positive operators I, Mathematica (Cluj) 9 (32), 77-83, (1967).
110. A. Lupaş, Some prorerties of the linear positive operators II, Mathematica (Cluj) 9 (32), 295-298, (1967).
111. I.I. Ibragimov and A.D. Gadžijev, On a sequence of linear positive operators, Dokl. Akad. Nauk SSSR 193, 1222-1225, (1970).
112. B. Wood, Graphic behaviour of positive linear operators, SIAM J. Appl. Math. 20, 329-335, (1971).
113. n-operators, Aequationes Math. (Cluj) 5, 19-37, (1970).
114. B. Della Vecchia, A new class of linear positive operators of Bernstein type, Anal. Numér. Théor. Approx. 17, 113-124, (1988).
115. G. Mastroianni and M.R. Occorsio, Una generalizzazione dell'operatore di Stancu, Rend. Acad. Sci. M.F.N. Ser. IV XLV, 495-511, (1978).
116. J.L. Durrmeyer, Une formule d'inversion de la transformée de Laplace. Applications á la théorie des moments, In Thése de 3e cycle, Fac. des Sciences de l'Université de Paris, Paris, (1967).
117. M.M. Derriennic, Sur l'approximation des fonctions d'une ou plusieurs variables par des polynômes de Bernstein modifiés et application au problème des moments, In Thése de 3e cycle, Université de Rennes, Rennes, (1978).
118. D. Andrica and C. Badea, Jensen's and Jessen's inequality, convexity-preserving and approximating polynomial operators and Korovkin's theorem, Research Seminars, Seminar on Math. Anal., (Edited by Faculty of Mathematics), Preprint No. 4, pp. 7-16, Babes-Bolyai University Cluj-Napoca, (1986).
119. Lj.M. Kocić, Some new results for convex functions with applications in the approximation theory, Ph.D. Thesis, Univ. of Niš, Niš, (1984).
120. Lj.M. Kocić and I.B. Lacković, Convexity criterion involving linear operators, Facta Univ. (Niš) 1, 13-22, (1986).
121. T.N.T. Goodman, Shape preserving representations, In Mathematical Methods in CAGD, (Edited by T. Lyche and L. Schumaker), pp. 333-351, Academic Press, Boston, (1989).
122. Lj.M. Kocić, On a conjecture concerning Bernstein polynomials, In Conf. Inst. of Mathematics, Physics and Mechanics, (Edited by Z. Bohte), pp. 67-71, Univ. of Ljubljana, Ljubljana, (1986).
123. Lj.M. Kocić, Extended shape preserving approximations, In Constructive Theory of Functions, (Edited by B. Sendov et al.), pp. 270-275, Bulgarian Academy of Sciences, Sofia, (1988).
124. J. Ponstein, Seven kinds of convexity, SIAM Rev. 9, 115-119, (1967).
125. V.A. Ubhaya, Uniform approximation by quasi-convex and convex functions, J. Approx. Theory 55, 326-336, (1988).
126. B.M. Brown, D. Elliot and D.F. Paget, Lipschitz constant for the Bernstein polynomials of a Lipschitz continuous functions, JAT 49, 196-199, (1987).
127. B. Della Vecchia, On the preservation of Lipschitz constants for some linear operators, Bolletino U. M. I. 7 (3-B), 125-136, (1989).
128. A.M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12, 1203-1215, (1962).
129. L. Lupaş, A property of the S.N. Bernstein operator, Mathematica (Cluj) 9 (32), 299-301, (1967).
130. L. Lupaş, On starshapedness preserving properties of a class of linear positive operators, Mathematica (Cluj) 12 (35), 105-109, (1970).
131. R. Bojanić and J. Roulier, Approximation of convex functions by convex splines and convexity-preserving continuous linear operators, Anal. Numér. Théor. Approx. 3, 143-150, (1974).
132. R.K. Beatson, Convex approximation by splines, SIAM J. Math. Anal. 12, 549-559, (1981).
133. S.L. Dodd and D.F. McAllister, Algorithms for computing shape preserving spline approximations to data, Numer. Math. 46, 159-174, (1985).
134. L.-E. Andersson and T. Elfving, Interpolation and approximation by monotone cubic splines, J. Approx. Theory 66, 302-333, (1991).
135. D. Girard and P.J. Laurent, Splines and estimation of nonlinear parameters, In Mathematical Methods in CAGD, (Edited by T. Lyche and L. Schumaker), pp. 273-298, Academic Press, Boston, (1989).
136. Lj.M. Kocić, Affine shape control of cubics, PU. M. A. Ser. B 3, 207-229, (1992).
137. R. A. DeVore, Monotone approximation by splines, SIAM J. Math. Anal. 8, 891-905, (1977).
138. R.A. DeVore, Monotone approximation by polynomials, SIAM J. Math. Anal. 8, 906-921, (1977).
139. G.G. Lorentz, Monotone approximation, In Inequalities, III, (Edited by O. Shisha), pp. 201-215, Acad. Press, New York, (1972).
140. G.G. Lorentz and K.L. Zeller, Degree of approximation by monotone polynomials I, J. Approx. Theory 1, 501-504, (1968).
141. D. Leviatan and H.N. Mhaskar, Comonotone approximation by splines of piecevise monotone functions, J. Approx. Theory 35, 364-369, (1982).
142. D. Leviatan, The degree of copositive approximation by polynomials, Proc. Amer. Math. Soc. 88, 101-105, (1983).
143. p, 0 < p < 1, for algebraic polynomials and rates of comonotone approximations, (in Russian), Mat. Zametki 25, 107-117, (1979).
144. A.S. Shvedov, Co-approximation rates of functions by algebraic polynomials, (in Russian), Mat. Zametki 29, 117-130, (1981).
145. A.S. Shvedov, Co-approximation by piecewise-monotone functions by polynomials, (in Russian), Mat. Zametki 30, 839-846, (1981).
146. R.K. Beatson, Restricted range of approximation by splines and variational inequalities, SIAM J. Numer. Anal. 19, 372-380, (1982).
147. R.K. Beatson and D. Leviatan, On comonotone approximation, Canad. Math. Bull. 26, 220-224, (1983).
148. J. Gilewicz and I.A. Shevchuk, Comonotone approximation, Techn. Report CPT-95/P.3203, Centre de Physique Théorique, Marseille, (May 1995).
149. G.A. Dzyubenko, J. Gilewicz and I.A. Shevchuk, Piecewise monotone pointwise approximation, Techn. Report CPT-94/P.3121, Centre de Physique Théorique, Marseille, (January 1995).
150. V.N. Konovalov and A.V. Pavlik, Positive, monotone and convex coapproximation of functions differentiable on a segment, In Preprint Series, (Edited by G.A. Dzyubenko et al.), Preprint 94.39, Inst. Math. NAS of Ukraine, Kyiv, (1994).
151. B. Lenze, On one-sided spline approximation operators, Numer. Func. Anal. Optim. 10, 167-180, (1989).
152. I.A. Shevchuk, Approximation of monotone functions by monotone polynomials, (in Russian), Mat. Sbornik 183, 63-78, (1992); English transi., Russian Acad. Sci. Sb. Math. 76, 51-64, (1993).
153. I.A. Shevchuk, Approximation of monotone functions by monotone polynomials, (in Russian), Mat. Sbornik 183, 63-78, (1992); English transi., Russian Acad. Sci. Sb. Math. 76, 51-64, (1993).
154. D. Legg and D. Townsend, Best monotone approximation in L∞[0, 1], J. Approx. Theory 42, 30-35, (1984).
155. D. Legg, Best monotone approximation in L∞[0, 1], Contemporary Math. 42, 105-111, (1985).
156. 1-approximation of bounded, approximately continuous functions on [0, 1] by nondecreasing functions, J. Approx. Theory 43, 178-189, (1985).
157. R.B. Darst and R. Houtari, Monotone approximation on an interval, Contemporary Math. 42, 43-44, (1985).
158. A.C. Calder and J.G. Laframboise, Multiple-water-bag simulation of inhomogeneous plasma motion near an electrode, J. Comput. Phys. 65, 18-45, (1986).
159. Laframboise and A.D. Stauffer, Optimum discrete approximation of the Maxwell distribution, AIAA J. 7, 520-523, (1969).
160. W. Gautschi, Discrete approximation to spherically symmetric distributions, Numer. Math. 44, 53-60, (1984).
161. W. Gautschi and G.V. Milovanović, Spline approximations to spherically symmetric distributions, Numer. Math. 49, 111-121, (1986).
162. M. Frontini, W. Gautschi and G.V. Milovanović, Moment-preserving spline approximation on finite intervals, Numer. Math. 50, 503-518, (1987).
163. M. Frontini and G.V. Milovanović, Moment-preserving spline approximation on finite intervals and Turán quadratures, Facta Univ. Ser. Math. Inform. 4, 45-56, (1989).
164. C.A. Micchelli, Monosplines and moment preserving spline approximation, In Numerical Integration III, (Edited by H. Brass and G. Hámmerlin), pp. 130-139, Birkhäuser, Basel, (1988).
165. G.V. Milovanović and M. A. Kovačević, Moment-preserving spline approximation and Turán quadratures, In Numerical Mathematics, (Singapore, 1988), (Edited by R.P. Agarwal, Y.M. Chow and S.J. Wilson), ISNM Vol. 86, pp. 357-365, Birkhäuser, Basel, (1988).